| L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s − 3·9-s + 10-s − 11-s + 2·13-s − 16-s − 6·17-s − 3·18-s + 19-s − 20-s − 22-s − 8·23-s + 25-s + 2·26-s − 6·29-s + 4·31-s + 5·32-s − 6·34-s + 3·36-s − 2·37-s + 38-s − 3·40-s − 10·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s − 9-s + 0.316·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.229·19-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 1/2·36-s − 0.328·37-s + 0.162·38-s − 0.474·40-s − 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.430298773506168656487379834590, −8.680265915687738043015894417363, −8.071416084937120999405606330586, −6.61497057113220838361920165158, −5.92455359343492777337204179724, −5.20480645886684182265145788830, −4.22208505788091945721791822553, −3.26824506410830084509707132514, −2.13131620331563136285075125502, 0,
2.13131620331563136285075125502, 3.26824506410830084509707132514, 4.22208505788091945721791822553, 5.20480645886684182265145788830, 5.92455359343492777337204179724, 6.61497057113220838361920165158, 8.071416084937120999405606330586, 8.680265915687738043015894417363, 9.430298773506168656487379834590
